Practice — Survival Analysis & Time-to-Event Modeling (6 questions)

A SaaS team tracks how many weeks eight trial users take to convert to a paid plan, with the data cut at week 12. Times are in weeks; "converted" is the event, and "still trialing" means the user was censored at that time (still in trial, or the trial expired and the user left, which the team treats as non-informative censoring).

1. Build the Kaplan-Meier table (at-risk, events, step factor, \hat S(t)) and read off the median time to conversion. Explain how the two users with a tied time of 5 and the two with a tied time of 8 are handled.
2. The analyst's colleague says: "Just compute the share who converted within 8 weeks: 4 converted by week 8 out of 8 users, so 50 percent." What is wrong, and what does the curve say instead?
3. Compute the Greenwood standard error of \hat S(8). Why would you report a log-log interval rather than \hat S\pm1.96\,\text{se}?
4. Another colleague proposes averaging the eight times (7.3 weeks) as "the mean time to convert" and dropping the three censored users to "keep only real conversions" (mean 6.6 weeks). Why are both wrong, and what summary would you report instead?

A data scientist fits a Cox model for subscription churn (time in months, event is cancellation) and presents the table below to the pricing team.

A Schoenfeld-residual test reports p=0.003 for annual_plan and p\gt0.4 for the others. The baseline survival at 12 months (all covariates at zero) is S_0(12)=0.60.

1. Explain what HR 0.45 for annual_plan means, and what it does NOT mean. A manager says "annual plans cut churn by 55 percent": give the correct 12-month churn for a user on an annual plan, with the other covariates at 0, and the relative churn reduction.
2. Interpret sessions_first_30d, and point out why this covariate is a problem for the claim "increase early engagement and churn drops 30 percent per 10 sessions".
3. What does the Schoenfeld result mean for the annual-plan row? List three ways to deal with it and what you would report to the pricing team meanwhile.
4. mobile_only has HR 0.95, p=0.35. State what you can and cannot conclude.